reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem
  SepQuadruples VERUM(A) = { [VERUM(A),0(A),{}.bound_QC-variables(A),id
  bound_QC-variables(A)] }
proof
  now
    let x be object;
    thus x in SepQuadruples VERUM(A) implies x =
     [VERUM(A),0(A),{}.bound_QC-variables(A),id bound_QC-variables(A)]
    proof
      assume
A1:   x in SepQuadruples VERUM(A);
      then consider q,t,K,f such that
A2:   x = [q,t,K,f] by DOMAIN_1:10;
A3:   now
        given x,v,h such that
        v++ = t and
        h +*({x} --> x.v) = f and
A4:     [All(x,q),v,K,h] in SepQuadruples VERUM(A) or [All(x,q),v,K\{.x
        .},h] in SepQuadruples VERUM(A);
        All(x,q) is_subformula_of VERUM(A) by A4,Th35;
        then All(x,q) = VERUM(A) by QC_LANG2:79;
        then VERUM(A) is universal by QC_LANG1:def 21;
        hence contradiction by QC_LANG1:20;
      end;
A5:   now
        given r,v such that
        t = v+QuantNbr r and
A6:     [r '&' q,v,K,f] in SepQuadruples VERUM(A);
        r '&' q is_subformula_of VERUM(A) by A6,Th35;
        then r '&' q = VERUM(A) by QC_LANG2:79;
        then VERUM(A) is conjunctive by QC_LANG1:def 20;
        hence contradiction by QC_LANG1:20;
      end;
A7:   now
        given r such that
A8:     [q '&' r, t, K,f] in SepQuadruples VERUM(A);
        q '&' r is_subformula_of VERUM(A) by A8,Th35;
        then q '&' r = VERUM(A) by QC_LANG2:79;
        then VERUM(A) is conjunctive by QC_LANG1:def 20;
        hence contradiction by QC_LANG1:20;
      end;
A9:   now
        assume ['not' q,t,K,f] in SepQuadruples VERUM(A);
        then 'not' q is_subformula_of VERUM(A) by Th35;
        then 'not' q = VERUM(A) by QC_LANG2:79;
        then VERUM(A) is negative by QC_LANG1:def 19;
        hence contradiction by QC_LANG1:20;
      end;
A10:     index VERUM(A) = 0(A) by Th22;
     set p = VERUM(A);
  [q,t,K,f] = [p,index p,{}.
bound_QC-variables(A),id bound_QC-variables(A)] or
   ['not' q,t,K,f]
in SepQuadruples p or
   (ex r st [q '&' r, t, K,f] in SepQuadruples p) or
  (ex r,u st t = u+QuantNbr r & [r '&' q,u,K,f] in SepQuadruples p) or
  ex x,u,h st u++ = t & h +*({x} --> x.u) = f & ([All(x,q),u,K,h]
  in SepQuadruples p or
     [All(x,q),u,K\{x},h] in SepQuadruples p) by A1,A2,Th34;
      hence x =
     [VERUM(A),0(A),{}.bound_QC-variables(A),id bound_QC-variables(A)]
       by A2,A7,A5,A3,A9,A10;
    end;
    index VERUM(A) = 0(A) by Th22;
    hence x = [VERUM(A),0(A),{}.bound_QC-variables(A),id bound_QC-variables(A)]
     implies x in SepQuadruples VERUM(A) by Th30;
  end;
  hence thesis by TARSKI:def 1;
end;
